PK 2018 Q17

Let the integers be \(\{2018, a_1, a_2, \ldots, a_n\}\). Their product is \(2018 \times a_1 \times a_2 \times \ldots \times a_n = 2018\) so we get \(a_1 \times a_2 \times \ldots \times a_n = 1\), meaning each \(a_i\) is either 1 or −1 (with an even number of occurrences of −1).
Their sum is \(2018 + a_1 + a_2 + \ldots + a_n = 2018\) so \(a_1 + a_2 + \ldots + a_n = 0\), meaning an equal number of occurrences of 1 and −1. Hence the list includes 2018 once, an even number of 1s and the same even number of −1s. Therefore the number of integers must be one more than a multiple of 4, so from the list of options only 2017 is possible.